How to calculate Cronbach's alpha for your thesis questionnaire, step by step

Cronbach's alpha is the number that appears in every thesis that uses a questionnaire. Yet many researchers report it without knowing exactly what it measures, and make mistakes an examiner spots at once: one alpha for the whole questionnaire but none for its constructs, negative items that were never reversed, or alpha offered as proof of validity. This article explains the coefficient, calculates it by hand on a small example, then shows how to read the results.

What is Cronbach's alpha? {#what}

Cronbach's alpha (α), published by Lee Cronbach in 1951, measures the internal consistency of a set of items: how far the answers to the items move together. If the items of a "behavioural engagement" construct really measure one idea, someone who strongly agrees with one tends to agree with the others, and someone who rejects one tends to reject the others.

It usually lies between 0 and 1, and the closer to 1 the higher the consistency. It can be negative, which is almost always a sign of an error in the data, such as a reverse item that was never reversed.

Remember that alpha is a measure of reliability, not validity: a construct can be very consistent while measuring something other than what you intended. The difference is explained in our article on questionnaire validity and reliability.

The formula {#formula}

α = (k ÷ (k − 1)) × (1 − sum of the item variances ÷ variance of the total score)

where k is the number of items in the construct. The idea: when the items are consistent, the variance of their sum is much larger than the sum of their separate variances, because the items rise and fall together. The ratio between the two approaches zero, and alpha approaches 1.

A worked example by hand {#example}

A construct of three items on a 5-point Likert scale, answered by five people:

Step 1: the variance of each item. We use the sample variance (dividing by n − 1 = 4).

• Item 1: mean 3.6, sum of squared deviations 5.2, so the variance = 5.2 ÷ 4 = 1.3
• Item 2: mean 4.0, sum of squared deviations 4.0, so the variance = 1.0
• Item 3: mean 3.6, sum of squared deviations 5.2, so the variance = 1.3

Sum of the item variances = 1.3 + 1.0 + 1.3 = 3.6

Step 2: the variance of the total score. The totals are 13, 8, 14, 8 and 13, with a mean of 11.2 and a sum of squared deviations of 34.8, so the variance = 34.8 ÷ 4 = 8.7

Step 3: put them into the formula.

α = (3 ÷ 2) × (1 − 3.6 ÷ 8.7) = 1.5 × (1 − 0.414) = 1.5 × 0.586 = 0.88

The result is 0.88: good reliability.

Note five respondents is far too few to compute reliability in a real study; the example is for illustration only, and its numbers are made up.

When is the value acceptable? {#thresholds}

The most widely quoted scale, from George & Mallery:

Three caveats worth knowing:

• 0.70 is a convention, not an absolute rule. Some fields and exploratory studies accept 0.60, and scales used to make decisions about individuals need more than 0.90.
• Alpha rises with the number of items. A 20-item construct can show a high alpha even when its items are only weakly related, and a 3-item construct can show a modest alpha while being sound.
• A very high value is not always good. Alpha above 0.95 can mean the items repeat each other in different words, lengthening the questionnaire without adding information.

Alpha if item deleted {#if-item-deleted}

This table recomputes alpha with each item left out in turn, to show which item weakens the construct. In our example:

Removing item 1 or 2 lowers alpha, so they support the construct. Removing item 3 raises it from 0.88 to 0.93: it is the least consistent with the others. Look at respondents B and E: the first gave it 2 and the other two items 3, the second gave it 5 and the other two 4, so it does not always move with the other two.

The table usually comes with the corrected item-total correlation: the item's correlation with the sum of the other items. A value below 0.30 often points to a weak item.

Important don't remove an item just because alpha would rise a little. If the item covers an important part of the concept, removing it weakens content validity. Remove one when the rise is clear and you find a reason in its wording or meaning, and say so in your thesis.

Common mistakes that lower reliability {#mistakes}

• Reverse items that were never reversed. The most common cause of a low or negative alpha. On a 5-point scale the new score is 6 minus the original. See our article on the Likert scale.
• Computing alpha only for the whole questionnaire. Alpha assumes the items measure one idea. Compute it for each construct, then for the whole instrument if your supervisor asks.
• Constructs of one or two items. A single item has no internal consistency to compute.
• A very small sample. The value swings a lot in small samples.
• A construct with no variation. If everyone gives the same answer, there is no consistency to measure.
• Missing answers you didn't notice. Most programs leave out of alpha anyone who didn't answer every item of the construct, so the sample shrinks without you noticing.

How to report it {#write-up}

Show a table with each construct, its number of items and its alpha, then the whole instrument, followed by a sentence that reads it:

"Cronbach's alpha for the instrument's constructs ranged from 0.81 to 0.92, and was 0.94 for the instrument as a whole, indicating high reliability for the purposes of the study."

If you removed items, say which and why, with alpha before and after.

Let Pulseform compute it on your questionnaire {#pulseform}

When you build your questionnaire in Pulseform as an academic survey, set the construct of each item and mark the reverse scored items, you don't need any of the arithmetic above:

• Cronbach's alpha for each construct with its verdict, from Excellent to Poor,
• Alpha if item deleted and the corrected item-total correlation, with the items whose removal improves alpha highlighted,
• Reliability within each group, such as men and women or each year of study,
• A clear message instead of a misleading number when the data can't support one, such as a single-item construct or a construct with no variation.

How to prepare the survey and read the results is in the academic surveys guide, and a full example with a construct that fails on purpose is in Read a report with me.

References {#sources}

The original sources of the formulas, thresholds and rules of thumb in this article:

• Cortina, J. M. (1993). What is coefficient alpha? An examination of theory and applications. Journal of Applied Psychology, 78(1), 98–104.
• Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297–334.
• George, D., & Mallery, P. (2003). SPSS for Windows step by step: A simple guide and reference, 11.0 update (4th ed.). Allyn & Bacon.
• Nunnally, J. C. (1978). Psychometric theory (2nd ed.). McGraw-Hill.
• Nunnally, J. C., & Bernstein, I. H. (1994). Psychometric theory (3rd ed.). McGraw-Hill.
• Streiner, D. L. (2003). Starting at the beginning: An introduction to coefficient alpha and internal consistency. Journal of Personality Assessment, 80(1), 99–103.